Risk and Return Statistics

Expected return, variance, covariance, and correlation of asset returns; historical evidence on equity vs bond returns; why diversification reduces risk without reducing expected return.

What we mean by “return”

A holding-period return on an asset over a single period is:

R = \frac{P_1 + D - P_0}{P_0}.

Where $P_0$ is purchase price, $P_1$ is sale price, $D$ is any dividend received during the period. Sometimes quoted as a percentage, sometimes as a decimal.

If we observe many such periods, we get a distribution of returns and can compute its statistics.

The five statistics that matter

For a single asset with possible returns $R_1, R_2, \ldots, R_n$ realized with probabilities $p_1, \ldots, p_n$ (or equally weighted across $n$ historical observations):

Expected return $E[R] = \sum p_i R_i$ . The mean.

Variance $\text{Var}(R) = E[(R - E[R])^2]$ . Average squared deviation from the mean.

Standard deviation $\sigma = \sqrt{\text{Var}(R)}$ . In the same units as return. The most common measure of “risk.”

For two assets $i$ and $j$ :

Covariance $\text{Cov}(R_i, R_j) = E[(R_i - E[R_i])(R_j - E[R_j])]$ . Positive when both tend to be above-mean together; negative when they offset.

Correlation $\rho_{ij} = \text{Cov}(R_i, R_j) / (\sigma_i \sigma_j)$ . Covariance normalized to $[-1, 1]$ .

Historical record (US, 1926-2024)

Compounded annual real returns (in 2024 dollars):

Asset class	Mean return	Std dev
Treasury bills (cash)	~0.5%	~3.5%
Treasury bonds (10y)	~2.5%	~10%
Corporate bonds (investment grade)	~3.5%	~9%
Large-cap stocks (S&P 500)	~7.5%	~17%
Small-cap stocks	~9%	~22%

The equity risk premium — the expected excess return of stocks over T-bills — has averaged 5-7% over the long run. It’s compensation for bearing the risk shown in the standard-deviation column.

A useful framing: the average annual return of US stocks is around 10% nominal. In any single year that average is almost never the realization — annual returns range from -40% to +50% in the historical record. The 10% is what you earn on average over decades, not next year.

S&P 500 nominal compound annual growth rate by decade. The ~10% long-run average hides enormous decade-to-decade variation: -0.9% per year through the 2000s, +19% per year through the 1950s and 1990s. The summary statistic is real; the path you actually experience depends heavily on when you started.Source: Compiled from S&P 500 total return series; standard pedagogical decadal compilation

Diversification: the only free lunch

Combine two assets with weights $w_1$ and $w_2 = 1 - w_1$ :

Portfolio expected return: $E[R_p] = w_1 E[R_1] + w_2 E[R_2]$ . (Weighted average; nothing special.)
Portfolio variance: $\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{12}$ .

That last term is the key. If $\rho_{12} < 1$ (the assets aren’t perfectly correlated), portfolio variance is less than the weighted average of individual variances. The lower the correlation, the bigger the variance reduction. At $\rho_{12} = -1$ , you can construct a portfolio with zero variance.

This is why owning 20 stocks is much safer than owning 1, even though each individual stock is just as risky as it was before. The diversifiable part of each stock’s risk cancels in the portfolio.

Systematic vs idiosyncratic risk

Decompose a stock’s return into two parts:

Systematic (market) risk — comes from things that affect all stocks (recession, interest rates, oil prices, geopolitics). Cannot be diversified away because every stock shares some exposure to it.
Idiosyncratic (firm-specific) risk — fires, lawsuits, CEO resignations, drug-trial failures. Can be diversified away by holding many stocks.

In a well-diversified portfolio, idiosyncratic risk averages to roughly zero (some firms have good idiosyncratic surprises, some bad — they cancel). Only systematic risk remains.

This decomposition matters for pricing. Investors should not be compensated for bearing risk they could have eliminated for free through diversification. So only systematic risk is priced. That’s the foundation for CAPM, two lessons from now.

A worked diversification example

Two stocks: A has $E[R_A] = 12\%$ , $\sigma_A = 30\%$ . B has $E[R_B] = 8\%$ , $\sigma_B = 20\%$ . Correlation $\rho_{AB} = 0.3$ .

50/50 portfolio:

$E[R_p] = 0.5 \times 12 + 0.5 \times 8 = 10\%$ (linear).
$\sigma_p^2 = 0.25 \times 900 + 0.25 \times 400 + 2 \times 0.25 \times 30 \times 20 \times 0.3 = 325$
$\sigma_p = \sqrt{325} \approx 18\%$ (less than 25%, the average).

You got an average return at less-than-average risk. That’s diversification at work.

If $\rho_{AB}$ had been 1.0 instead of 0.3:

$\sigma_p = 0.5 \times 30 + 0.5 \times 20 = 25\%$ .

No diversification benefit when correlations are perfect.

Why this matters

Every risk-pricing model in the rest of this course assumes investors hold diversified portfolios and care only about systematic risk. That assumption is grounded in this lesson’s math. If investors really held only one stock (no diversification), the right discount rate would include compensation for that stock’s total variance — much higher than what CAPM prescribes. Empirically, investors do diversify (most retail investors via index funds; institutional investors via mandate); the assumption is reasonable.

Next lesson: the systematic framework for choosing which portfolio to hold.